Theorem signslema 28783

Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

Hypotheses

Ref	Expression
signslema.1	|- ( ph -> E e. NN0 )
signslema.2	|- ( ph -> F e. NN0 )
signslema.3	|- ( ph -> G e. NN0 )
signslema.4	|- ( ph -> H e. NN0 )
signslema.5	|- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
signslema.6	|- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )

Assertion

Ref	Expression
signslema	|- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Proof of Theorem signslema

Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 |- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
2	1	simpld 457	. . . . 5 |- ( ph -> E < G )
3	2	adantr 463	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> E < G )
4		signslema.4	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
5	4	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> H e. CC )
6		signslema.2	. . . . . . . . . 10 |- ( ph -> F e. NN0 )
7	6	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> F e. CC )
8	5, 7	subcld 9922	. . . . . . . 8 |- ( ph -> ( H - F ) e. CC )
9		signslema.3	. . . . . . . . . 10 |- ( ph -> G e. NN0 )
10	9	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> G e. CC )
11		signslema.1	. . . . . . . . . 10 |- ( ph -> E e. NN0 )
12	11	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> E e. CC )
13	10, 12	subcld 9922	. . . . . . . 8 |- ( ph -> ( G - E ) e. CC )
14	8, 13	subeq0ad 9932	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 <-> ( H - F ) = ( G - E ) ) )
15	14	biimpa 482	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( H - F ) = ( G - E ) )
16	15	breq2d 4451	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 0 < ( H - F ) <-> 0 < ( G - E ) ) )
17	6	nn0red 10849	. . . . . . 7 |- ( ph -> F e. RR )
18	4	nn0red 10849	. . . . . . 7 |- ( ph -> H e. RR )
19	17, 18	posdifd 10135	. . . . . 6 |- ( ph -> ( F < H <-> 0 < ( H - F ) ) )
20	19	adantr 463	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( F < H <-> 0 < ( H - F ) ) )
21	11	nn0red 10849	. . . . . . 7 |- ( ph -> E e. RR )
22	9	nn0red 10849	. . . . . . 7 |- ( ph -> G e. RR )
23	21, 22	posdifd 10135	. . . . . 6 |- ( ph -> ( E < G <-> 0 < ( G - E ) ) )
24	23	adantr 463	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> 0 < ( G - E ) ) )
25	16, 20, 24	3bitr4rd 286	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> F < H ) )
26	3, 25	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> F < H )
27		0red 9586	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 e. RR )
28	22, 21	resubcld 9983	. . . . . 6 |- ( ph -> ( G - E ) e. RR )
29	28	adantr 463	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) e. RR )
30	18, 17	resubcld 9983	. . . . . 6 |- ( ph -> ( H - F ) e. RR )
31	30	adantr 463	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( H - F ) e. RR )
32	2	adantr 463	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> E < G )
33	23	adantr 463	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( E < G <-> 0 < ( G - E ) ) )
34	32, 33	mpbid 210	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( G - E ) )
35		2pos 10623	. . . . . . 7 |- 0 < 2
36		breq2 4443	. . . . . . 7 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> ( 0 < ( ( H - F ) - ( G - E ) ) <-> 0 < 2 ) )
37	35, 36	mpbiri 233	. . . . . 6 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> 0 < ( ( H - F ) - ( G - E ) ) )
38	28, 30	posdifd 10135	. . . . . . 7 |- ( ph -> ( ( G - E ) < ( H - F ) <-> 0 < ( ( H - F ) - ( G - E ) ) ) )
39	38	biimpar 483	. . . . . 6 |- ( ( ph /\ 0 < ( ( H - F ) - ( G - E ) ) ) -> ( G - E ) < ( H - F ) )
40	37, 39	sylan2 472	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) < ( H - F ) )
41	27, 29, 31, 34, 40	lttrd 9732	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( H - F ) )
42	19	adantr 463	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( F < H <-> 0 < ( H - F ) ) )
43	41, 42	mpbird 232	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> F < H )
44	5, 10, 7, 12	sub4d 9971	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) = ( ( H - F ) - ( G - E ) ) )
45		signslema.6	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )
46	44, 45	eqeltrrd 2543	. . . 4 |- ( ph -> ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } )
47		ovex 6298	. . . . 5 |- ( ( H - F ) - ( G - E ) ) e. _V
48	47	elpr 4034	. . . 4 |- ( ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } <-> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
49	46, 48	sylib 196	. . 3 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
50	26, 43, 49	mpjaodan 784	. 2 |- ( ph -> F < H )
51	1	simprd 461	. . . . 5 |- ( ph -> -. 2 || ( G - E ) )
52	51	adantr 463	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( G - E ) )
53	15	breq2d 4451	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 2 || ( H - F ) <-> 2 || ( G - E ) ) )
54	52, 53	mtbird 299	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( H - F ) )
55		2z 10892	. . . . . . 7 |- 2 e. ZZ
56	9	nn0zd 10963	. . . . . . . 8 |- ( ph -> G e. ZZ )
57	11	nn0zd 10963	. . . . . . . 8 |- ( ph -> E e. ZZ )
58	56, 57	zsubcld 10970	. . . . . . 7 |- ( ph -> ( G - E ) e. ZZ )
59		dvdsaddr 14109	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( G - E ) e. ZZ ) -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
60	55, 58, 59	sylancr 661	. . . . . 6 |- ( ph -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
61	51, 60	mtbid 298	. . . . 5 |- ( ph -> -. 2 || ( ( G - E ) + 2 ) )
62	61	adantr 463	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( ( G - E ) + 2 ) )
63		2cnd 10604	. . . . . . 7 |- ( ph -> 2 e. CC )
64	8, 13, 63	subaddd 9940	. . . . . 6 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 2 <-> ( ( G - E ) + 2 ) = ( H - F ) ) )
65	64	biimpa 482	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( ( G - E ) + 2 ) = ( H - F ) )
66	65	breq2d 4451	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( 2 || ( ( G - E ) + 2 ) <-> 2 || ( H - F ) ) )
67	62, 66	mtbid 298	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( H - F ) )
68	54, 67, 49	mpjaodan 784	. 2 |- ( ph -> -. 2 || ( H - F ) )
69	50, 68	jca 530	1 |- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   {cpr 4018   class class class wbr 4439  (class class class)co 6270   RRcr 9480   0cc0 9481   + caddc 9484   < clt 9617   - cmin 9796   2c2 10581   NN0cn0 10791   ZZcz 10860   || cdvds 14070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-dvds 14071

This theorem is referenced by: (None)